This course will cover broadly the many aspects of constructing a model predictive controller for a real application: model formulation, constraints, integrating disturbance models, moving horizon state estimation, setpoint tracking with unreachable setpoints, and offset-free control with nonzero disturbances. The course will present in addition the theoretical analysis tools that are used to establish the closed-loop control properties and the statistical properties of moving horizon estimation: Lyapunov stability theory, the theory of random variables, and conditional probability.

Students will learn fundamental principles of quantum mechanics, master mathematical details of atomic wavefunctions (especially for Hydrogen-like atoms), understand quantum mechanical treatment of angular momentum and spin, and know how to obtain atomic term symbols.

Introduction to variational methods; many-electron systems; semi-empirical methods; perturbation theory; computational methods.

Discrete Optimization II is a generic name for COMM 618; a more precise name for this rendition is something like "Connections between Discrete Optimization and Machine Learning". The purpose of this class is three-fold:

1. Introduce some common discrete optimization problems, e.g., the knapsack problem, the Traveling Salesman Problem, and network flow. In addition, we will discuss algorithms for these problems.
2. Discuss tools and techniques in machine learning, e.g., reinforcement learning and deep learning.
3. Explore research papers that either (i) use machine learning tools to solve discrete optimization problems or (ii) use discrete optimization tools to address questions in machine learning.

In light of purpose 3, the course will have a few meetings that are more focused on the discrete optimization (respectively, the machine learning) followed by a few meetings focused on the machine learning (respectively, the discrete optimization). When possible, the lectures will stress the mathematics behind these concepts. We will likely not analyze algorithms as thoroughly as a course dedicated to algorithms nor will we likely analyze machine learning tools as thoroughly as a course dedicated to machine learning.

CPSC 424 is an advanced computer graphics course focussed on geometric modeling, i.e. creation and manipulation of shapes. The topics covered in this course are: Introduction to curves and surfaces, in particular splines, subdivision surfaces, polygonal meshes. Principles and mathematical foundations for representing complex geometry for computer graphics and numerical simulations. Practical applications of different modeling techniques.

A brief outline: formulation and analysis of algorithms for sparse matrix computations; direct and iterative solvers for large and sparse linear systems arising from discretization of elliptic partial differential equations, constrained optimization problems, and other problems; numerical solution of sparse eigenvalue problems; iterative methods for sparse least-squares problems; applications in fluid dynamics, electromagnetics, multiphysics, graphs and networks; other topics, time permitting

We introduce basic principles and techniques in the fields of data mining and machine learning. These are some of the key tools behind the emerging field of data science. These techniques are now running behind the scenes to discover patterns and make predictions in various applications in our daily lives. We will focus on many of the core data mining and machine learning technlogies, with motivating applications from a variety of disciplines.

We study the design of algorithms for combinatorial optimization problems typically omitted from undergraduate programs. Our approach is largely based on using tools from linear programming and convexity. Example problems we study are weighted matchings, detecting negative cycles in undirected graphs, T-joins, virtual private network design, and Travelling Salesman Problem.

This course introduces fundamental principles and methods for optimization, which plays a central role in a variety of engineering problems. Our main focus is on convex programs, a class of optimization problems with a special, yet commonly-encountered structure, that everyone who uses computational mathematics will benefit from knowing about.

Specifically, the course is designed to give the graduate student thorough knowledge about how to formulate, recognize, solve and interpret the solution of convex programs. Representative list of topics includes: convexity, first/second -order optimality conditions, linear/quadratic/cone programs, duality and KKT conditions, first/second -order methods, interior point methods, ADMM. General concepts will be illustrated through applications in machine learning, statistics and signal processing.

Students entering the class should have a solid background in linear algebra and basic real analysis. A working knowledge of basic statistics and probability is also encouraged, although not necessary.

Greens functions are analytic maps from the data for a linear differential equation problem to the solution. We will explore techniques for finding Greens functions for different types of problems and the insight they give to the properties of the models they represent. We will also investigate some calculus of variations problems and the analysis of the finite element method as examples of variational principles.

Review of basic ideas of numerical analysis: Newton's method, interpolation, and quadrature. Spatial discretization with Finite Different, Finite Volume, Spectral, and Finite Element Methods. Time stepping. Course mark based on challenging assignments and a project.

Students will learn high-dimensional probability and how it applies to dimension reduction, learning theory, and compressed sensing. The latter will be used as a running example.

The class will include (the juiciest parts of):
1. Tail bounds for high-dimensional functions with random input (concentration of measure);
2. Non-asymptotic random matrix theory;
3. Extrema of random processes.

In this course we formulate and analyze continuum-based mathematical models of phenomena in a wide range of areas of application. A list of the topics for the course is given below. The
mathematical modeling typically leads to PDE/ODE models, free and moving boundary value
problems, etc. For each topic covered I plan to give some insight into the formulation of appropriate
mathematical models from the underlying physics. Wherever possible, asymptotic analysis is then
used to systematically simplify the underlying model into a more tractable form. Asymptotic and
numerical methods, based on MATLAB, will then be used to provide insight into the solution
behavior. Finally, for each topic considered, we will examine a recent journal article (either in an
Applied Math or Engineering journal) that is closely related to the topic.)

Topics to be covered:
 Materials Science Modeling: MEMS modeling, theroelastic contact
 Lubrication theory and slow viscous flow phenomena; singularity formation
 Bifurcation problems in combustion theory and nonlinear heat transfer
 Nonlinear Oscillators: Forced Oscillators, Entrainment, Stick-Slip Oscillations
 Dynamical Hysteresis for ODE’s; Slow passage problems
 Floquet Theory: stability of periodic solutions
 Delay-Differential Equations; Machine-Tool Vibrations, Car-Following Models
 Solidification theory, moving boundary problems (as time permits)

MATH 462/560 provides a broad overview of Mathematical Biology at an introductory level. The scope is obviously subject to the limitations of time and instructor knowledge and interests - this is a HUGE area of research.

It is intended for early stage math bio grad students or senior undergrads thinking of heading in that direction, general applied math grad students interested in finding out more about biology applications, and grad students in other related departments interested in getting some mathematical and computational modelling experience.

The course is organized around a sample of topics in biology that have seen a significant amount of mathematical modelling over the years. Currently, I'm including content from ecology, evolution and evolutionary game theory, epidemiology, biochemistry and gene regulation, cell biology, electrophysiology and developmental biology. However, this list changes gradually from year to year, to reflect students' and my own interests. The mathematical modelling methods and techniques covered are those that typically arise in the biological applications listed above. For example, I will cover models using ordinary and partial differential equations, stochastic processes, agent-based models and introduce techniques from bifurcation theory, asymptotics, dimensional analysis, numerical solution methods, and parameter estimation. An emphasis will be placed on reading and discussing classic and current papers.

We will study a variety of modelling approaches for epidemics, as well as within host dynamics of pathogens and multi scale models. You will leave the class with a good understanding of the following topics:

* Some basics of pathogens and spread mechanisms (virus, bacteria, prion, macroparasites)
* Stochastic (branching process) models for epidemics and within host pathogen dynamics
* SIR and its differential-equation cousins for epidemics and within host pathogen dynamics
* A certain amount about practical aspects of measurement, model parameterization and making projections
* Multi-scale models for within- and between-host dynamics
* Concepts and models for pathogen competition and evolution
* Basics of the immune system and some modelling paradigms
* Lots of new covid-related examples

Prerequisites: The main prerequisites are fascination with mathematical modelling and interest in infectious disease. In previous years students with undergraduate degrees in Statistics, Physics, Mathematics, Bioinformatics/Systems Bio and Engineering have succeeded in the class. Technically, second to third year mathematics (ODEs, some basic dynamical systems, basic linear algebra) and ability to set up and analyze models on the computer (i.e. Matlab, R, Python…). Usually most of the students are at the graduate level but I have had interested upper year undergraduates who had no problems. The material is not super technical on the mathematical side, and in fact, the more technical nuggets are much less important than the bigger concepts.

Nuts and bolts: class will run in person on Wednesday and Friday, 9am-10.20am, in Geography 201. First class will be on Weds, Sept 7.

There will be a number of homework assignments, students will present papers from the literature, and (in small groups) complete a short original modelling project with a write up and presentation.

Questions? Please get in touch: coombs@math.ubc.ca. Or just show up at the first class.

The course covers modeling techniques (ODEs, PDEs), from elementary to more advanced, and highlights some classical models for cell structures, cell motility, pattern formation, and single and collective cell behaviour. Communication between biologists and mathematicians is emphasized. Multiscale computation of cell dynamics using the freeware Morpheus is taught and used to explore models of current biological phenomena.

This is a research-oriented graduate course designed to introduce tensors (or multi-dimensional arrays) and their uses in statistics, machine learning, and the sciences. We will illustrate fundamental theoretical properties of several types of tensor decompositions, including CP-decomposition, nonnegative matrix and tensor decomposition, Tucker decomposition as well as tensor network decompositions arising from physics. We will see how these naturally come up in hidden variable models, Gaussian mixture models, directed and undirected graphical models, blind source separation, independent component analysis, and quantum physics. We will discuss algorithms for computing such decompositions, and will exhibit open problems.

This the same as the undergraduate course MATH 405 Numerical Methods for Differential Equations. As the title suggests, it is a first course on numerical methods for differential equations. We will cover theory, algorithms and code (Julia).

Continuum description of solids, Deformation Gradient and Stress and Strain measures, Equilibrium Equations, Linear Elasticity problems in small deformations, Constitutive Equations: Isotropic and Anisotropic Linear Elastic Materials, Resolution Strategies for Elasticity problems, Wave propagation and high strain rate testing of materials.

https://mech-modsim.sites.olt.ubc.ca/teaching/mech-561-elasticity/

Selected advanced topics in CFD, typically chosen from: Finite volume methods on curvilinear meshes and structured mesh generation. Finite volume methods on unstructured meshes. Adjoint methods. Reynolds-averaged form of the Navier-Stokes equations and turbulence modelling. Three-dimensional flows. Compressible flows.